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Implementing the Quantum von Neumann Architecture with Superconducting Circuits Matteo Mariantoni1;4;x, H. Wang1;∗, T. Yamamoto1;2, M. Neeley1;y, Radoslaw C. Bialczak1, Y. Chen1, M. Lenander1, Erik Lucero1, A. D. O’Connell1, D. Sank1, M. Weides1;z, J. Wenner1, Y. Yin1, J. Zhao1, A. N. Korotkov3, A. N. Cleland1;4, and John M. Martinis1;4;x 1Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA 2Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan 3Department of Electrical Engineering, University of California, Riverside, CA 92521, USA 4California NanoSystems Institute, University of California, Santa Barbara, CA 93106-9530, USA ∗Present address: Department of Physics, Zhejiang University, Hangzhou 310027, China. yPresent address: Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, MA 02420-9108, USA. zPresent address: National Institute of Standards and Technology, Boulder, CO 80305, USA. xTo whom correspondence should be addressed. E-mail: [email protected] (M. M.); [email protected] (J. M. M.) last updated: September 16, 2011 The von Neumann architecture for a classical computer comprises a central processing unit and a memory holding instructions and data. We demonstrate a quantum central processing unit that exchanges data with a quantum random-access memory integrated on a chip, with instructions stored on a classical computer. We test our quantum machine by executing codes that involve seven quantum elements: Two superconduct- ing qubits coupled through a quantum bus, two quantum memories, and two zeroing registers. Two vital algorithms for quantum computing are demonstrated, the quantum Fourier transform, with 66 % process fidelity, and the three-qubit Toffoli OR phase gate, with 98 % phase fidelity. Our results, in combination especially with longer qubit coherence, illustrate a potentially viable approach to factoring numbers and implementing simple quantum error correction codes. Quantum processors 1–4 based on nuclear magnetic resonance 5–7, trapped ions 8–10, and semiconducting de- vices 11 were used to realize Shor’s quantum factoring algorithm 5 and quantum error correction 6,8. The quantum operations underlying these algorithms include two-qubit gates 2,3, the quantum Fourier transform 7,9, and three- qubit Toffoli gates 10,12. In addition to a quantum processor, a second critical element for a quantum machine is a quantum memory, which has been demonstrated, e.g., using optical systems to map photonic entanglement into and out of atomic ensembles 13. Superconducting quantum circuits 14 have met a number of milestones, including demonstrations of two-qubit gates 5,15–17,19,20 and the advanced control of both qubit and photonic quantum states 7,19,20,22. We demonstrate a superconducting integrated circuit that combines a processor, executing the quantum Fourier transform and a three-qubit Toffoli-class OR gate, with a memory and a zeroing register in a single device. This combination of a quantum central processing unit (quCPU) and a quantum random-access memory (quRAM), which comprise two key elements of a classical von Neumann architecture, defines our quantum von Neumann architecture. In our architecture (Fig. 1A), the quCPU performs one-, two-, and three-qubit gates that process quantum information, and the adjacent quRAM allows quantum information to be written, read out, and zeroed. The 5,7,19,22 quCPU includes two superconducting phase qubits Q1 and Q2, connected through a coupling bus provided 1 Figure 1: The quantum von Neumann architecture. (A) The quCPU (blue box) includes two qubits Q1 and Q2 and the bus resonator B. The quRAM (magenta boxes) comprises two memories M1 and M2 and two zeroing registers Z1 and Z2. The horizontal dotted lines indicate connections between computational elements. The vertical direction represents frequency, where the memory and zeroing registers are fixed 7 in frequency, while the qubit transition frequencies can be tuned via z-pulses (grey dashed double arrows). (B) Swap spectroscopy for Q1 (left) and Q2 (right): Qubit excited state jei probability Pe (color scale) vs. z-pulse amplitude (vertical axis) and delay time ∆τ (horizontal axis), after exciting the qubit with a π-pulse. At zero z-pulse amplitude the qubits are at their idle points, where they have an energy relaxation time Trel ' 400 ns. A separate Ramsey experiment yields the qubits’ dephasing time Tdeph ' 200 ns. By tuning the z-pulse amplitude, the qubit transition frequencies f and f can be varied between ' 5:5 and 8 GHz. For z-pulse amplitudes indicated by B and M for Q , and Q1 Q2 1 1 7 by B and M2 for Q2, the “chevron pattern” of a qubit-resonator interaction is observed . The transition frequencies of B, M1, and M2 are f = 6:82 GHz, f = 6:29 GHz, and f = 6:34 GHz, respectively. From the chevron oscillation we obtain the qubit-resonator coupling B M1 M2 p strengths, which for both the resonator bus and the memories are ' 20 MHz (splitting) for the jgi $ jei qubit transition, and ≈ 2 faster for 22 the jei $ jfi transition (jgi, jei, and jfi are the three lowest qubit states) . For all resonators Trel ' 4 µs. Swap spectroscopy also reveals that the qubits interact with several modes associated with spurious two-level systems. Two of them, Z1 and Z2, are used as zeroing registers. Their transition frequencies are f = 6:08 GHz and f = 7:51 GHz, respectively, with coupling strength to the qubits of ' 17 MHz. Z1 Z2 by a superconducting microwave resonator B. The quRAM comprises two superconducting resonators M1 and M2 that serve as quantum memories, as well as a pair of zeroing registers Z1 and Z2, two-level systems that are used to dump quantum information. The chip geometry is similar to that in Refs. 7,22, with the addition of the two zeroing registers. Figure 1B shows the characterization of the device by means of swap spectroscopy 7. The computational capability of our architecture is displayed in Fig. 2A, where a 7-channel quantum circuit, yielding a 128 dimensional Hilbert space, executes a prototypical algorithm. First, we create a Bell state between p 22 Q1 and Q2 using a series of π-pulse, iSWAP, and iSWAP operations (step I, a to c) . The corresponding density matrix ρ^(I) [Fig. 2C (I)] is measured by quantum state tomography. The Bell state is then written into the quantum 22 memories M1 and M2 by an iSWAP pulse (step II) , leaving the qubits in their ground state jgi, with density matrix ρ^(II) [Fig. 2C (II)]. While storing the first Bell state in M1 and M2, a second Bell state with density matrix 2 Figure 2: Programming the quantum von Neumann architecture. (A) Quantum algorithm comprising 7 independent channels interacting through five computational steps. Dotted and solid lines represent channels inp the ground and excited/superposition states, respectively. A black rectangle represents a π-pulse; two crosses connected by a solid line a iSWAP; an open and a closed circle connected by a single arrow an iSWAP; oblique arrows indicate decay from a zeroing register. (B) Calibration of the zeroing gates. Each qubit is prepared in jei, interacts on resonance with its zeroing register for a time τz, and its probability Pe measured, with Pe plotted vs. τz (large and small blue circles). The solid green line is a decaying cosine fit to the data. The black arrows indicate the zeroing time for each qubit. (C) Density matrices ρ^(I); ρ^(II);:::; ρ^(V) of the Q1-Q2 state for each step in A (scale key on bottom left). Grey arrows: Ideal state. Red and black arrows and black dots: Measured state (black arrows indicate errors). The off-diagonal elements of ρ^(I), ρ^(III), and ρ^(V) have different angles because 26 of dynamic phases . Fidelities: F(I) = 0:772 ± 0:003, F(II) = 0:916 ± 0:002, F(III) = 0:689 ± 0:003, F(IV) = 0:913 ± 0:002, and F(V) = 0:606 ± 0:003. Concurrences: C(I) = 0:593 ± 0:006, C(II) = 0:029 ± 0:005, C(III) = 0:436 ± 0:007, C(IV) = 0:019 ± 0:005, and C(V) = 0:345 ± 0:008.(D) Comparison of fidelity F as a function of storage time τst for a Bell state stored in Q1 and Q2 (blue circles) vs. that stored in M1 and M2 (magenta squares; error bars smaller than symbols). The solid lines are exponential fits to data. (E) As in D, but for the concurrence C. In D and E the vertical black dotted line indicates the time delay (' 59 ns) associated with memory storage, with respect to storage in the qubits, due to the writing and reading operations (II) and (V) in A. ρ^(III) [Fig. 2C (III)] is created between the qubits, using a sequence similar to the first operation (step III, a to c). In order to re-use the qubits Q1 and Q2, for example to read out the quantum information stored in the mem- 23 ories M1 and M2, the second Bell state has to be dumped . This is accomplished using two zeroing gates, by bringing Q1 on resonance with Z1 and Q2 with Z2 for a zeroing time τz, corresponding to a full iSWAP (step IV). Figure 2B shows the corresponding dynamics, where each qubit, initially in the excited state jei, is measured in the ground state jgi after ' 30 ns. The density matrix ρ^(IV) of the zeroed two-qubit system is shown in Fig. 2C (IV). Once zeroed, the qubits can be used to read the memories (step V), allowing us to verify that, at the end of the algorithm, the stored state is still entangled. This is clearly demonstrated by the density matrix shown in Fig.

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